Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails

Reducing Livestock Effects on Public Lands in the Western United States as the Climate Changes: A Reply to Svejcar et al.

Beschta et al. (2013) synthesized the ecological effects 41 of climate change and ungulate grazing on western public lands, grounding their recommendations in ecological considerations and federal agency legal authority and obligations. Svejcar et al. (2014) suggest that Beschta et al. (2013) neither “present a balanced synthesis of the scientific literature” nor “reflect the complexities associated with herbivore grazing.” Svejcar et al. (2014) “dispute the notion that eliminating [livestock] grazing will provide a solution to problems created by climate change,” although we made no such claim. Instead, Beschta et al. (2013: p. 474) indicate that removal or reduction of livestock across large areas of public land will reduce a pervasive ecological stress, diminishing cumulative impacts on these ecosystems under climate change. We respond to three livestock grazing issues raised by Svejcar et al. (2014): (1) legacy vs. contemporary effects, (2) fuels reduction and fire effects, and (3) grazing complexity and restoration.Keywords: Climate change, Public lands, Restoration, Ungulates, Livestock grazing, Biodiversit